1. Technical Field of the Invention
This invention generally relates to a computer implemented physical signal processing method, apparatus and article of manufacture. More particularly, this invention relates to a computer implemented method and apparatus for processing two-dimensional nonlinear, nonstationary physical signals.
2. Description of Related Art
Analyzing physical signals is a difficult problem confronting many industries. These industries have harnessed various computer implemented methods to process data taken from physical phenomena such as earthquakes, ocean waves, tsunamis, ocean surface elevation and wind. Unfortunately, previous methods have not yielded results which are physically meaningful.
Among the difficulties is that representing physical processes with physical signals may present one or more of the following problems:
(a) The total data span is too short; PA1 (b) The data are nonstationary; and PA1 (c) The data represent nonlinear processes.
Although problems (a)-(c) are separate issues, the first two problems are related because a data section shorter than the longest time scale of a stationary process can appear to be nonstationary. Because many physical events are transient, the data representative of those events are nonstationary. For example, a transient event such as an earthquake will produce nonstationary data when measured. Nevertheless, the nonstationary character of such data is ignored or the effects assumed to be negligible. This assumption may lead to inaccurate results and incorrect interpretation of the underlying physics as explained below.
A variety of techniques have been applied to nonlinear, nonstationary physical signals. For example, many computer implemented methods apply Fourier spectral analysis to examine the energy-frequency distribution of such signals.
Although the Fourier transform that is applied by these computer implemented methods is valid under extremely general conditions, there are some crucial restrictions: the system must be linear, and the data must be strictly periodic or stationary. If these conditions are not met, then the resulting spectrum will not make sense physically.
A common technique for meeting the linearity condition is to approximate the physical phenomena with at least one linear system. Although linear approximation is an adequate solution for some applications, many physical phenomena are highly nonlinear and do not admit a reasonably accurate linear approximation.
Furthermore, imperfect probes/sensors and numerical schemes may contaminate data representative of the phenomenon. For example, the interactions of imperfect probes with a perfect linear system can make the final data nonlinear.
Many recorded physical signals are of finite duration, nonstationary, and nonlinear because they are derived from physical processes that are nonlinear either intrinsically or through interactions with imperfect probes or numerical schemes. Under these conditions, computer implemented methods which apples Fourier spectral analysis are of limited use. For lack of alternatives, however, such methods still apply Fourier spectral analysis to process such data.
Moreover, the indiscriminate use of Fourier spectral analysis in these methods and the adoption of the stationary and linear assumptions may give misleading results some of which are described below.
First, the Fourier spectrum defines uniform harmonic components globally. Therefore, the Fourier spectrum needs many additional harmonic components to simulate nonstationary data that are nonuniform globally. As a result, energy is spread over a wide frequency range.
For example, using a delta function to represent the flash of light from a lightning bolt will give a phase-locked wide white Fourier spectrum. Here, many Fourier components are added to simulate the nonstationary nature of the data in the time domain, but their existence diverts energy to a much wider frequency domain. Constrained by the conservation of energy principle, these spurious harmonics and the wide frequency spectrum cannot faithfully represent the true energy density of the lighting in the frequency and time space.
More seriously, the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give the final delta function representative of the lightning. Thus, the Fourier components might make mathematical sense, but they often do not make physical sense when applied.
Although no physical process can be represented exactly by a delta function, some physical data such as the near field strong earthquake energy signals are of extremely short duration. Such earthquake energy signals almost approach a delta function, and they always give artificially wide Fourier spectra.
Second, Fourier spectral analysis uses a linear superposition of trigonometric functions to represent the data. Therefore, additional harmonic components are required to simulate deformed wave profiles. Such deformations, as will be shown later, are the direct consequence of nonlinear effects. Whenever the form of the data deviates from a pure sine or cosine function, the Fourier spectrum will contain harmonics.
Furthermore, both nonstationarity and nonlinearity can induce spurious harmonic components that cause unwanted energy spreading and artificial frequency smearing in the Fourier spectrum. The consequence is incorrect interpretation of physical phenomenon due to the misleading energy-frequency distribution for nonlinear and nonstationary data representing the physical phenomenon.
According to the above background, the state of the art does not provide a useful computer implemented tool for analyzing nonlinear, nonstationary physical signals. Geophysical signals provide a good example of a class of signals in which this invention is applicable. Grandparent application Ser. No. 08/872,586 filed on Jun. 10, 1997, U.S. Pat. No. 5,983,162, illustrates several types of nonlinear, nonstationary geophysical signals which are very difficult to analyze with traditional computer implemented techniques including earthquake signals, water wave signals, tsunami signals, ocean altitude and ocean circulation signals.
This application extends the techniques of the parent and grandparent applications to the processing of two-dimensional signals. Two-dimensional signals such as images of physical phenomena are an increasing subject of various processing techniques. In fact, the above-described prior art techniques such as Fourier analysis are routinely applied to process two-dimensional signals.
Many two-dimensional signals are nonlinear and/or nonstationary. Therefore, like the one-dimensional data processing techniques described above, the conventional processing techniques are simply inadequate to process such two-dimensional data.
Moreover, two-dimensional data analysis methods are utilized to separate the various scales contained in the data. For example, in image processing, scale separation is a necessary step toward an objective. analysis of the information contents. However, many images contain inhomogeneous scales which cannot be satisfactorily separated, analyzed or processed by conventional image processing methods.